4.2 Motion and gravitational radiation 4 Strong Gravity and Gravitational 4 Strong Gravity and Gravitational

4.1 Strong-field systems in general relativity 

4.1.1 Defining weak and strong gravity 

In the solar system, gravity is weak, in the sense that the Newtonian gravitational potential and related variables (tex2html_wrap_inline4963) are much smaller than unity everywhere. This is the basis for the post-Newtonian expansion and for the ``parametrized post-Newtonian'' framework described in Sec.  3.2 . ``Strong-field'' systems are those for which the simple 1PN approximation of the PPN framework is no longer appropriate. This can occur in a number of situations:

Of course, some systems cannot be properly described by any post-Newtonian approximation because their behavior is fundamentally controlled by strong gravity. These include the imploding cores of supernovae, the final merger of two compact objects, the quasinormal-mode vibrations of neutron stars and black holes, the structure of rapidly rotating neutron stars, and so on. Phenomena such as these must be analysed using different techniques. Chief among these is the full solution of Einstein's equations via numerical methods. This field of ``numerical relativity'' is a rapidly growing and maturing branch of gravitational physics, whose description is beyond the scope of this article. Another is black hole perturbation theory (see [93Jump To The Next Citation Point In The Article] for a review).

4.1.2 Compact bodies and the strong equivalence principle 

When dealing with the motion and gravitational wave generation by orbiting bodies, one finds a remarkable simplification within general relativity. As long as the bodies are sufficiently well-separated that one can ignore tidal interactions and other effects that depend upon the finite extent of the bodies (such as their quadrupole and higher multipole moments), then all aspects of their orbital behavior and gravitational wave generation can be characterized by just two parameters: mass and angular momentum. Whether their internal structure is highly relativistic, as in black holes or neutron stars, or non-relativistic as in the Earth and Sun, only the mass and angular momentum are needed. Furthermore, both quantities are measurable in principle by examining the external gravitational field of the bodies, and make no reference whatsoever to their interiors.

Damour [39Jump To The Next Citation Point In The Article] calls this the ``effacement'' of the bodies' internal structure. It is a consequence of the strong equivalence principle (SEP), described in Section  3.1.2 .

General relativity satisfies SEP because it contains one and only one gravitational field, the spacetime metric tex2html_wrap_inline4263 . Consider the motion of a body in a binary system, whose size is small compared to the binary separation. Surround the body by a region that is large compared to the size of the body, yet small compared to the separation. Because of the general covariance of the theory, one can choose a freely-falling coordinate system which comoves with the body, whose spacetime metric takes the Minkowski form at its outer boundary (ignoring tidal effects generated by the companion). There is thus no evidence of the presence of the companion body, and the structure of the chosen body can be obtained using the field equations of GR in this coordinate system. Far from the chosen body, the metric is characterized by the mass and angular momentum (assuming that one ignores quadrupole and higher multipole moments of the body) as measured far from the body using orbiting test particles and gyroscopes. These asymptotically measured quantities are oblivious to the body's internal structure. A black hole of mass m and a planet of mass m would produce identical spacetimes in this outer region.

The geometry of this region surrounding the one body must be matched to the geometry provided by the companion body. Einstein's equations provide consistency conditions for this matching that yield constraints on the motion of the bodies. These are the equations of motion. As a result the motion of two planets of mass and angular momentum tex2html_wrap_inline4899, tex2html_wrap_inline4991, tex2html_wrap_inline4993 and tex2html_wrap_inline4995 is identical to that of two black holes of the same mass and angular momentum (again, ignoring tidal effects).

This effacement does not occur in an alternative gravitional theory like scalar-tensor gravity. There, in addition to the spacetime metric, a scalar field tex2html_wrap_inline4521 is generated by the masses of the bodies, and controls the local value of the gravitational coupling constant (i.e. G is a function of tex2html_wrap_inline4521). Now, in the local frame surrounding one of the bodies in our binary system, while the metric can still be made Minkowskian far away, the scalar field will take on a value tex2html_wrap_inline4519 determined by the companion body. This can affect the value of G inside the chosen body, alter its internal structure (specifically its gravitational binding energy) and hence alter its mass. Effectively, each mass becomes several functions tex2html_wrap_inline5007 of the value of the scalar field at its location, and several distinct masses come into play, inertial mass, gravitational mass, ``radiation'' mass, etc. The precise nature of the functions will depend on the body, specifically on its gravitational binding energy, and as a result, the motion and gravitational radiation may depend on the internal structure of each body. For compact bodies such as neutron stars, and black holes these internal structure effects could be large; for example, the gravitational binding energy of a neutron star can be 40 percent of its total mass. At 1PN order, the leading manifestation of this effect is the Nordtvedt effect.

This is how the study of orbiting systems containing compact objects provides strong-field tests of general relativity. Even though the strong-field nature of the bodies is effaced in GR, it is not in other theories, thus any result in agreement with the predictions of GR constitutes a kind of ``null'' test of strong-field gravity.

4.2 Motion and gravitational radiation 4 Strong Gravity and Gravitational 4 Strong Gravity and Gravitational

image The Confrontation between General Relativity and Experiment
Clifford M. Will
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de